1. Field of the Invention
The present invention relates to a data converting apparatus defined by an appropriate number of discrete value coefficients and, more specifically, to a method and an apparatus for determining coefficients which, in response to given input data, permit an output with a minimum of errors relative to ideal output data. The invention applies illustratively to techniques for determining coefficients of a color converting apparatus on a hierarchical neural network having discrete value coefficients as well as for determining representative points for color conversion by use of a representative point table and interpolation.
2. Description of the Related Art
For a data converting apparatus defined by a given number of discrete value coefficients, the variety of values each coefficient may take is limited if the tolerable range of coefficients is limited. Thus if all combinations of coefficients are evaluated for error, at least one optimal coefficient exists among them. This technique of searching through all spaces of coefficients yields optimal coefficients unfailingly. Generally, however, the possible combinations of coefficients are so numerous that it is not realistic to search through their spaces for coefficient calculation. It is thus common practice to compute coefficients by use of deductive methods or various approximating methods based on the characteristics of each specific data converting apparatus.
For some data converting apparatuses containing discrete value coefficients, optimal coefficients may be determined in response to a given input/output correspondence. In such cases, conventional deductive methods may be used to determine the coefficients with no problem.
In other cases, coefficients may be determined deductively but do not take discrete values. For example, a data converting apparatus for calculating determinants made of discrete value coefficients is capable of deductively computing each coefficient having a minimum error but yields a result other than a discrete value for each coefficient. Fractions of such results must be rounded to the nearest whole number, counted as a whole number, or disregarded. The rounding operation turns each coefficient into a value different from an optimum value, which leads to worsening errors.
Similar rounding operations also apply to determinants of which the effective precision is limited, e.g., where fixed-point product-sum operations with finite accuracy are carried out. The data converting apparatus performing the operations is subject to a further decline in precision because of the rounding carried out during the calculating process. Such arrangements would require taking into consideration the characteristics of the rounding operation in the computing process in order to obtain optimal coefficients. However, such consideration has yet to be taken up conventionally because of very complicated relations that exist between coefficients and errors. So far, precision has been boosted by establishing finer steps of effective accuracy and of coefficients in the calculating process.
In a data converting apparatus using a representative point table and interpolation, converted data are recorded beforehand to those lattice points in the table which are composed of representative points of each input dimension. The recorded data are interpolated so as to yield a result of the conversion. For that data converting apparatus, the coefficients written in the table are generally constituted by data in effect following the conversion of representative points. Although errors generally result from interpolation, there has been no conventional technique for optimizing the coefficients recorded in the lattice points while concurrently taking into account conversion errors of inputs made through other than lattice points. Conventionally, the level of precision has been enhanced by increasing the number of representative points in the input dimensions.
There exist data converting apparatuses of which the coefficients are continuous and which partially differentiate errors using each coefficient. Such apparatuses are run under a known scheme which determines coefficients through successive approximation of differential equations. On neural networks, in particular, coefficient calculating methods represented by the back propagation method are generally used.
Where discrete value coefficients are involved, errors cannot be partially differentiated by such coefficients. That is, the above methods represented by the back propagation method cannot be utilized.
The difficulty above has been bypassed conventionally, as long as the data converting apparatus in question has the same structure as outlined above and is capable of partial differentiation, by a method whereby coefficients are calculated in advance using continuous values and the results are rounded so as to acquire discrete value coefficients. This method, as with the above-mentioned setup using determinants, is also plagued by the worsening precision attributable to rounding operations.
Learning by hardware has conventionally been based on techniques similar to back propagation wherein like differential equations are approximated with finite accuracy. Such techniques have the disadvantage of the learning process getting halted when the error of back propagation drops below the calculated level of precision.
Neural networks based on fixed-point arithmetic of finite accuracy, as with fixed-point determinants, are also affected by the rounding during calculations. Relations between coefficients and errors on such networks become so complicated that it is practically impossible to obtain optimum discrete values. Conventionally, precision has been enhanced usually by establishing finer steps of effective accuracy and of coefficients in the calculating process. To set up the finely stepped accuracy levels and coefficients involves complicating the structure of the data converting apparatus in question.
The method above concerning neural networks requires that differential equations be solved through successive approximation. The requirement further complicates the structure of the apparatus based on that method.
As far as neural networks are concerned, Japanese Published Patent Application No. Hei 4-282476 (corresponding to U.S. Pat. No. 5,459,817) discloses a learning method of a relatively simple structure. The disclosed method allows update differences of coefficients to be randomly generated so that the coefficients are each updated by alternatively selecting the old or the updated value thereof. The method requires providing a feature for generating random numbers for each synapse. Because only signs are controlled using random numbers with no control exerted on the update quantities, no control is effected on the update quantities from the earliest stage of learning up to an advanced stage of successive learning. This poses two problems: the progress of learning tends to be slow, and the result can be minimally converged if the alternative selection procedure is not structured appropriately.
Of the above-mentioned methods, all except the one involving searching through all spaces of coefficients differ in the way coefficients are computed in accordance with the structure of the data converting apparatus in question. If any one of such methods is adopted as the basis for constituting a data converting apparatus, the structure of that apparatus becomes complicated. As mentioned earlier, the apparatus based on the method for searching through all spaces of coefficients is simply structured but not practical because it needs to search through huge spaces.
Where it is desired to improve the level of accuracy for a conventional data converting apparatus, the structure of the apparatus and that of the method for use therewith need to be changed.
Where coefficients are to be updated on the basis of differential coefficients, it is mandatory to devise suitable differential equations. For such cases as canceling which cannot be dealt with by differentiation, differential equations cannot be devised and thus coefficients cannot be determined.